Geometric Variational Problems and Rearrangement Inequalities

AbstractAward：DMS-0308040主要研究者：Almut Burchard提出了两组问题，一组与弹性曲线有关，另一组与重排不等式有关。 第一组问题是关于三维空间中运动的弹性曲线的动力学，在由其曲率决定的力的作用下，并受其不可伸展的约束。主要问题是，从光滑的初始数据开始的解是否存在于所有时间，或者它们是否在有限时间内变得奇异。 对于无穷曲线的长时间演化，孤立波有望起到重要作用。 张力的边值问题将被详细研究，目标是获得关于曲线的位置和速度的张力的精确边界。在第二组问题中，PI提出解决经典库仑能量不等式的一个定量版本。此外，我们还研究了重积分的Brascamp-Lieb-Luttinger不等式的不定性情形. 此外，PI参与了两个跨学科的合作，统计网络微积分及其在数据通信中的应用（与计算机科学和系统工程的同事），以及极化细胞内膜上蛋白质的聚类（与生物学的同事）。上述弹性曲线方程提供了一类具有几何力和几何约束的非线性波动方程的简单例子。在这个问题中遇到的分析困难，以及用于克服它们的技术，预计将与更复杂的几何力力学系统有关。 虽然这项工作是几何动机，弹性线的动力学显然与机械工程中的结构问题有关。 这里讨论的基本适定性问题可能对数值解方案的收敛性有影响。 重排不等式是最小化非线性非凸泛函的少数几何工具之一。 拟议的工作力求加强这些工具，并扩大其适用性。 其意义包括对称星系构型的稳定性和其他动力学稳定性问题。 应用合作的一个显著贡献是，它们使生物学和工程学的研究人员接触到了简单而有趣的数学的日常使用。 网络演算的结果集中在接纳控制和大型数据网络的服务保证问题。 这项生物学工作对蛋白质分选机制有着重要意义，而蛋白质分选机制对细胞的基本功能至关重要。

AbstractAward: DMS-0308040Principal Investigator: Almut BurchardTwo sets of problems are proposed, one related with elasticcurves, and the other with rearrangement inequalities. The firstset of problems concerns the dynamics of an elastic curve movingin three-dimensional space, under forces determined by itscurvature and subject to the constraint that it is inextensible.The main question is whether solutions starting from sufficientlysmooth initial data exist for all time, or whether they maybecome singular in finite time. For the long-time evolution ofan infinite curve, solitary waves are expected to play animportant role. The boundary value problem for the tension willbe studied in detail, with the goal of obtaining sharp bounds forthe tension in terms of the position and velocity of the curve.In the second set of problems, the PI proposes to settle aconjectured quantitative version of the classical rearrangementinequality for the Coulomb energy. Furthermore, the cases ofequality in the Brascamp-Lieb-Luttinger inequality for multipleintegrals will be investigated. In addition, the PI participatesin two interdisciplinary collaborations, on Statistical NetworkCalculus and its applications to data traffic (with colleaguesfrom Computer Science and Systems Engineering), and on theclustering of proteins on membranes within polarized cells (withcolleagues from Biology).The elastic curve equations described above provide a simpleexample of a class of nonlinear wave equations with geometricforces and geometric constraints. The analytic difficultiesencountered in this problem, and the techniques used to overcomethem, are expected to be relevant for more complex mechanicalsystems with geometric forces. While the work is geometricallymotivated, the dynamics of elastic wires are clearly relevant tostructural questions in Mechanical Engineering. The basicwell-posedness questions addressed here could have implicationsfor the convergence of numerical solution schemes. Rearrangementinequalities are among the few geometric tools for minimizingnonlinear, non-convex functionals. The proposed work strives tosharpen those tools and extend their applicability. Implicationsinclude the stability of symmetric galaxy configurations andother dynamical stability problems. A notable contribution ofthe applied collaborations is that they expose researchers inBiology and Engineering to the everyday use of simple, butrigorous Mathematics. The Network Calculus results center onquestions of admission control and service guarantees in largedata networks. The biological work has implication for themechanics of protein sorting, which is crucial to the function ofthe cell at a basic level.